Mizohata–Takeuchi estimates in the plane

Abstract

AbstractSuppose that is a smooth compact hypersurface in and is an appropriate measure on . If is the extension operator associated with , then the Mizohata–Takeuchi conjecture asserts that for all functions and weights , where the is taken over all tubes in of cross‐section 1, and . This paper investigates how far we can go in proving the Mizohata–Takeuchi conjecture in if we only take the decay properties of into consideration. As a consequence of our results, we obtain new estimates for a class of convex curves that include exponentially flat ones such as , , .